Almost everywhere and norm convergence of Approximate Identity and Fej\'er means of trigonometric and Vilenkin systems

TL;DR

This paper establishes almost everywhere and norm convergence of general approximation kernels, including Fejér means, for trigonometric and Vilenkin systems, using novel methods to address kernel property differences.

Contribution

It introduces a unified approach to prove convergence of approximation methods for both systems, overcoming limitations of traditional techniques.

Findings

Proved almost everywhere convergence for a broad class of kernels.

Established norm convergence of summability methods for trigonometric systems.

Developed alternative methods for a.e. convergence of Fejér means in Vilenkin systems.

Abstract

In this paper, we investigate very general approximation kernels with special properties, called an approximate identity, and prove almost everywhere and norm convergence of these general methods, which consists of a class of summability methods and provide norm and a.e. convergence of these summability methods with respect to the trigonometric system. Investigations of these summations can be used to obtain norm convergence of Fej\'er means with respect to the Vilenkin system also, but these methods are not useful to study a.e. convergence in this case, because of some special properties of the kernels of Fej\'er means. Despite these different properties we give alternative methods to prove almost everywhere convergence of Fej\'er means with respect to the Vilenkin systems.